Here is how my text defines self-similarity:
We call $M \subset \mathbb R^d$ self-similar if there are $T_1, \ldots, T_m \subsetneqq M$ and similarity maps $\alpha_1, \ldots, \alpha_m$ such that $\cup_{i=1}^m T_i =M$ and for all $i=1,\ldots,m$ holds $\alpha_i(T_i)=M.$
I understand that if we allow the amount of subsets to be uncountably infinite then every set is self-similar, which is no good.
But I don't quite see what would happed if we allow it to be countable, that is a sequence of subsets, each similar to the whole set. What would be an example of an unwanted set becoming self-similar in this case?
Given an open set $M$, you could take, for each rational point inside $M$ (that is, each point, all of whose coordinates are rational), a small copy of $M$ containing that point in such a way that the union would be all of $M$. And there are only countably many rational points.